Estimates for the Szegö kernel on a model non-pseudoconvex domain (Q938820)

In the present paper, the Szegö kernel on a domain in \({\mathbb C}^2\) of the type of a model domain \[ \Omega:= \{ (z_1,z_2)\in {\mathbb C}^2\,\,| \,\, \Im (z_2) + b (\Re z_1) <0\} \] is studied for a certain twice differentiable function \(b:{\mathbb R}\to {\mathbb R}\) that exhibits polynomial growth. It is known that \(\Omega\) is pseudoconvex if and only if \(b''\geq 0\), meaning that \(b\) is convex. The points on \(\partial \Omega\) are denoted as \((z_1,z_2)=(x,y,t, -b(x)) \equiv (x,y,t)\). For the special (non-convex) function defined by \(b(x) := (x-1)^2\) for \(x \geq 1/2\) , \(b(x):= \frac{1}{2} -x^2\), if \(x^2 \leq \frac{1}{4}\), and \(b(x) := (x+1)^2\), for \(x \leq -1/2\) detailed estimates of the Szegö kernel are established in the article. They show that, contrary to the case, where \(b\) is convex, the Szegö kernel \(S((x,y,t),\,(r,s,u)\,)\) shows singularities also at certain points off the diagonal.